Device, method, and medium for predicting a probability of an occurrence of a data

ABSTRACT

In a Bayes mixture probability density calculator for calculating Bayes mixture probability density which reduces a logarithmic loss A modified Bayes mixture probability density is calculated by mixing traditional Bayes mixture probability density calculated on given model S with a small part of Bayes mixture probability density for exponential fiber bundle on the S. Likewise, a prediction probability density calculator is configured by including the Bayes mixture probability density calculator, and by using Jeffreys prior distribution in traditional Bayes procedure on the S.

BACKGROUND OF THE INVENTION

[0001] This invention relates to technology for statistical predictionand, in particular, to technology for prediction based on Bayesprocedure.

[0002] Conventionally, a wide variety of methods have been proposed tostatistically predict a data on the basis of a sequence of datagenerated from the unknown source. Among the methods, Bayes predictionprocedure has been widely known and has been described or explained invarious textbooks concerned with statistics and so forth.

[0003] As a problem to be solved by such statistical prediction, thereis a problem for sequentially predicting, by the use of an estimationresult, next data which appear after the data sequence. As regards thisproblem, proof has been made about the fact that a specific Bayesprocedure exhibits a very good minimax property by using a particularprior distribution which may be referred to as Jeffreys priordistribution. Such a specific Bayes procedure will be called Jefferyprocedure hereinafter. This proof is done by B. Clarke and A. R. Barronin an article which is published in Journal of Statistical Planning andInference, 41:37-60, 1994, and which is entitled “Jeffreys prior isasymptotically least favorable under entropy risk”. This procedure isguaranteed to be always optimum whenever a probability distributionhypothesis class is assumed to be a general smooth model class, althoughsome mathematical restrictions are required in strict sense.

[0004] Herein, let logarithmic regret be used as another index. In thisevent also, it is again proved that the Jeffery procedure has a minimaxproperty on the assumption that a probability distribution hypothesisclass belongs to an exponential family. This proof is made by J.Takeuchi and A. R. Barron in a paper entitled “Asymptotically minimaxregret for exponential families”, in Proceedings of 20th Symposium onInformation Theory and Its Applications, pp. 665-668, 1997.

[0005] Furthermore, the problem of the sequential prediction can bereplaced by a problem which provides a joint (or simultaneous)probability distribution of a data sequence obtained by cumulativelymultiplying prediction probability distributions.

[0006] These proofs suggest that the Jeffreys procedure can haveexcellent performance except that the prediction problem is sequential,when the performance measure is the logarithmic loss.

[0007] Thus, it has been proved by Clarke and Barron and by Takeuchi andBarron that the Bayes procedure is effective when the Jeffreys priordistribution is used. However, the Bayes procedure is effective onlywhen the model class of the probability distribution is restricted tothe exponential family which is very unique, in the case where theperformance measure is the logarithmic regret instead of redundancy.

[0008] Under the circumstances, it is assumed that the probabilitydistribution model class belongs to a general smooth model class whichis different from the exponential family. In this case, the Jeffreysprocedure described in above B. Clarke and A. R. Barron's document doesnot guarantee the minimax property. To the contrary, it is confirmed bythe instant inventors in this case that the Jeffreys procedure does nothave the minimax property.

[0009] Furthermore, it often happens that a similar reduction ofperformance takes place in a general Bayes procedure different from theJeffreys procedure when estimation is made by using the logarithmicregret in lieu of the redundancy.

SUMMARY OF THE INVENTION

[0010] It is an object of this invention to provide a method which iscapable of preventing a reduction of performance.

[0011] It is a specific object of this invention to provide improvedJeffreys procedure which can accomplish a minimax property even whenlogarithmic regret is used a performance measure instead of redundancy.

[0012] According to a first embodiment of the invention, a Bayes mixturedensity calculator operable in response to a sequence of vectorsx^(n)=(x₁, x₂, . . . , x_(n)) selected from a vector value set χ toproduce a Bayes mixture density on occurrence of the x^(n), comprising aprobability density calculator, supplied with a sequence of data x^(t)and a vector value parameter u, for calculating a probability densityfor the x^(t), p(x^(t)|u), a Bayes mixture calculator for calculating afirst approximation value of a Bayes mixture density p_(w)(x^(n)) on thebasis of a prior distribution w(u) predetermined by the probabilitydensity calculator to produce the first approximation value, an enlargedmixture calculator for calculating a second approximation value of aBayes mixture m(x^(n)) on exponential fiber bundle in cooperation withthe probability density calculator to produce the second approximationvalue, and a whole mixture calculator for calculating(1−ε)p_(w)(x^(n))+ε·m(x^(n)) to produce a calculation result by mixingthe first approximation value of the Bayes mixture density p_(w)(x^(n))with a part of the second approximation value of the Bayes mixturem(x^(n)) at a rate of 1−ε:ε to produce the calculation result where ε isa value smaller than unity.

[0013] According to a second embodiment of the invention which can bemodified based on the first embodiment of the invention, a Jeffreysmixture density calculator operable in response to a sequence of vectorx^(n)=(x₁, x₂, . . . , x_(n)) selected from a vector value set χ toproduce a Bayes mixture density on occurrence of the x^(n), comprising aprobability density calculator responsive to a sequence of data x^(t)and a vector value parameter u for calculating a probability densityp(x^(t)|u) for the x^(t), a Jeffreys mixture calculator for calculatinga first approximation value of a Bayes mixture density p_(J)(x^(n))based on a Jeffreys prior distribution w_(J)(u) in cooperation with theprobability density calculator to produce the first approximation value,an enlarged mixture calculator for calculating a second approximationvalue of a Bayes mixture m(x^(n)) on exponential fiber bundle incooperation with the probability density calculator to produce thesecond approximation value, and a whole mixture calculator forcalculating (1−ε)p_(J)(x^(n))+ε·m(x^(n)) to produce a calculation resultby mixing the first approximation value of the Bayes mixture densityp_(J)(x^(n)) with a part of the second approximation value of the Bayesmixture m(x^(n)) at a rate of 1−ε:ε to produce the calculation resultwhere ε is a value smaller than unity.

[0014] Also, when hypothesis class is curved exponential family, it ispossible to provide with a third embodiment of the invention bymodifying the first embodiment of the invention. According to the thirdembodiment of the invention, a Bayes mixture density calculator operablein response to a sequence of vector x^(n)=(x₁,x₂, . . . ,x_(n)) selectedfrom a vector value set χ to produce a Bayes mixture density onoccurrence of the x^(n), comprising a probability density calculatorresponsive to a sequence of data x^(t) and a vector value parameter ufor outputting probability density p(x^(t)|u) for the x^(t) on curvedexponential family, a Bayes mixture calculator for calculating a firstapproximation value of a Bayes mixture density p_(w)(x^(n)) on the basisof a prior distribution w(u) predetermined by the probability densitycalculator to produce the first approximation value, an enlarged mixturecalculator for calculating a second approximation value of a Bayesmixture m(x^(n)) on exponential family including curved exponentialfamily in cooperation with the probability density calculator to producethe second approximation value, and a whole mixture calculator forcalculating (1−ε)p_(w)(x^(n))+ε·m(x^(n)) to produce a calculation resultby mixing the first approximation value of the Bayes mixture densityp_(w)(x^(n)) with a part of the second approximation value of the Bayesmixture m(x^(n)) at a rate of 1−ε:ε to produce the calculation resultwhere ε is a value smaller than unity.

[0015] According to a forth embodiment of the invention which can bemodified based on the third embodiment of the invention, a Jeffreysmixture density calculator operable in response to a sequence of vectorx^(n)=(x₁, x₂, . . . , x_(n)) selected from a vector value set χ toproduce a Bayes mixture density on occurrence of the x^(n), comprising aprobability density calculator responsive to a sequence of data x^(t)and a vector value parameter u for calculating probability densityp(x^(t)|u) for the x^(t) on curved exponential family, a Jeffreysmixture calculator for calculating a first approximation value of aBayes mixture density p_(J)(x^(n)) based on a Jeffreys priordistribution w_(J)(u) in cooperation with the probability densitycalculator to produce the first approximation value, an enlarged mixturecalculator for calculating a second approximation value of a Bayesmixture m(x^(n)) on exponential family including curved exponentialfamily in cooperation with the probability density calculator to producethe second approximation value, and a whole mixture calculator forcalculating (1−ε)p_(J)(x^(n))+ε·m(x^(n)) to produce a calculation resultby mixing the first approximation value of the Bayes mixture densityp_(J)(x^(n)) with a part of the second approximation value of the Bayesmixture m(x^(n)) at a ratio of 1−ε:ε to produce the calculation resultwhere ε is a value smaller than unity.

[0016] According to a fifth embodiment of the invention, a predictionprobability density calculator operable in response to a sequence ofvector x^(n)=(x₁, x₂, . . . , x_(n)) selected from a vector value set χand x_(n+1) to produce a prediction probability density on occurrence ofthe x_(n+1), comprising a joint probability calculator structured by theBayes mixture density calculator claimed in claim 1 for calculating amodified Bayes mixture density q^((ε))(x^(n)) and q^((ε))(x^(n+1)) basedon predetermined prior distribution to produce first calculation resultsand a divider responsive to the calculation results for calculatingprobability density q^((ε))(x^(n+1))/q^((ε))(x^(n)) produce a secondcalculation result with the first calculation results kept intact.

[0017] According to a sixth embodiment of the invention which can bemodified based on the fifth embodiment of the invention, a predictionprobability density calculator operable in response to a sequence ofvector x^(n)=(x₁, x₂, . . . , x_(n)) selected from a vector value set χand x_(n+1) to produce a prediction probability density on occurrence ofthe x_(n+1), comprising a joint probability calculator structured by theJeffreys mixture density calculator claimed in claim 2 for calculating amodified Jeffreys mixture density q^((ε))(x^(n)) and q^((ε))(x^(n+1)) toproduce first calculation results and a divider response to thecalculation results for calculating a probability densityq^((ε))(x^(n+1))/q^((ε))(x^(n)) to produce a second calculation resultwith the first calculation results kept intact.

[0018] Also, when hypothesis class is curved exponential family, it ispossible to provide with a seventh embodiment of the invention bymodifying the fifth embodiment of the invention. According the seventhembodiment of the invention, a prediction probability density calculatoroperable in response to a sequence of vector x^(n)=(x₁, x₂, . . . ,x_(n)) selected from a vector value set χ and x_(n+1) to produce aprediction probability density on occurrence of the x_(n+1), comprisinga joint probability calculator structured by the Bayes mixture densitycalculator claimed in claim 3 for calculating a modified Bayes mixturedensity q^((ε))(x^(n)) and q^((ε))(x^(n+1)) based on a predeterminedprior distribution to produce first calculation results and a dividerresponsive to the calculation results for calculating a probabilitydensity q^((ε))(x^(n+1))/q^((ε)(x) ^(n)) to produce a second calculationresult with the first calculation results kept intact.

[0019] According to an eighth embodiment of the invention which can bemodified based on the seventh embodiment of the invention, a predictionprobability density calculator operable in response to a sequence ofvector x^(n)=(x₁, x₂, . . . , x_(n)) selected from a vector value set χand x_(n+1) to produce a prediction probability density on occurrence ofthe x_(n+1), comprising a joint probability calculator structured by theJeffreys mixture density probability calculator claimed in claim 4 forcalculating a modified Jeffreys mixture density q^((ε))(x^(n)) andq^((ε))(x^(n+1)) to produce first calculation results and a dividerresponsive to the calculation results for calculating a probabilitydensity q^((ε))(x^(n+1))/q^((ε))(x^(n)) to produce a second calculationresult with the first calculation results kept intact.

BRIEF DESCRIPTION OF THE DRAWING

[0020]FIG. 1 shows a block diagram for use in describing a methodaccording to a first embodiment of the invention, which is executed bythe use of a first modified Bayes mixture distribution calculator;

[0021]FIG. 2 shows a block diagram for use in describing a methodaccording to a second embodiment of the invention, which is executed bythe use of a first modified Jeffreys mixture distribution calculator;

[0022]FIG. 3 shows a block diagram for use in describing a methodaccording to a third embodiment of the invention, which is executed bythe use of a second modified Bayes mixture distribution calculator;

[0023]FIG. 4 shows a block diagram for use in describing a methodaccording to a fourth embodiment of the invention, which is executed bythe use of a second modified Jeffreys mixture distribution calculator;

[0024]FIG. 5 shows a prediction probability calculator used in a methodaccording to fifth trough eighth embodiments of the invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0025] First, explanation is made about symbols used in thisspecification. Let ν be a σ-finite measure on the Borel subsets ofk-dimensional euclidean space

^(k) and χ be the support of ν. For example, it is assumed that Lebesguemeasure dx on the

^(d) is ν(dx) and

^(k) itself is χ (conversely, more general measure space could beassumed).

[0026] Herein, let consideration be made about a problem or a procedurewhich calculates, for each natural number t, a probability density ofx_(t+1) in response to a sequence x^(t) and x_(t+1). Here,$x^{t}\overset{def}{=}{( {x_{1},x_{2},\ldots \quad,x_{t}} ) \in \chi^{t}}$

[0027] x^(t) ^(def) =(x₁, x₂, . . . , x_(t))∈χ^(t)

[0028] and x_(t+1)∈χ. Such a procedure is assumed to be expressed asq(x_(t+1)|x^(t)). This means that a probability density of x_(t+1) isexpressed on condition that x^(t) are given. In this event, thefollowing equation holds.∫_(χ)  q(x_(t + 1)x^(t))v(  x_(t + 1)) = 1

[0029] In the above equation, q(x_(t+1)|x^(t)) is referred to asprediction probability distribution for t+1-th data x_(t+1).

[0030] Then, if${q( x^{n} )}\overset{def}{=}{\prod\limits_{t = 0}^{n - 1}{q( {x_{t + 1}x^{t}} )}}$

[0031] (assuming that q(x₁) is defined even if t=0), the followingequation holds. ∫_(χ^(n))  q(x^(n))v(  x^(n)) = 1${where},\quad {{v( {x^{n}} )}\overset{def}{=}{\prod\limits_{t = 1}^{n}{{v( {x^{t}} )}.}}}$

[0032] Therefore, q defines a joint probability distribution on infinitesequence set χ^(∞) (that is, q defines stochastic process). Giving astochastic process q, a prediction procedure is determined.

[0033] Next, a model class is determined. Let p(x|u) be a probabilitydensity of x∈χ based on a measure ν, where u is a real-valued parameterof d-dimensional. Then, the model class is defined by;$S\overset{def}{=}\{ {{p( {\cdot {u}} )}:{u \in U}} \}$

[0034] This class may be referred to as a hypothesis class. Where, let Ube a subset of

^(d). Assuming that p(x|u) is differentiable twice for u. And when let Kbe compact set included in U, S(K) is given by:

[0035]${S(K)}\overset{def}{=}\{ {{p( {\cdot {u}} )}:{u \in K}} \}$

[0036] Furthermore, definition is made as follows.${p( {x^{n}u} )}\overset{def}{=}{\prod\limits_{t = 1}^{n}{p( {x_{t}u} )}}$

[0037] That is, for a sequence of data x^(n), assuming that each elementx_(t) independently follows the same distribution p(·|u) (each elementis specified by i.i.d. which is an abbreviation of an independentlyidentical distributed state). For simplicity, in the specification, suchassumption is introduced. However, the method according to the inventionmay be easily expanded to the case where the each element x_(t) is noti.i.d.

[0038] A prior distribution is defined as a probability distribution inwhich a parameter u is regarded as a random variable. It is presumedthat density for Lebesgue measure du of a certain prior distribution isprovided as w(n). Then, p_(w) is considered as probability density onχ^(n), if it is given by:

[0039]${p_{w}( x^{n} )}\overset{def}{=}{\int{{p( {x^{n}u} )}{w(u)}\quad {u}}}$

[0040] Thus obtained p_(w) is referred to as Bayes mixture with priordensity w.

[0041] Next, definition of Jeffreys prior distribution is recalled.Fisher information of parameter u is represented as J(u). That is, ijcomponent of d-dimensional square matrix is obtained by followingequation.${J_{ij}(u)} = {- {E_{u}\lbrack \frac{{\partial^{2}\log}\quad {p( {xu} )}}{{\partial u_{j}}{\partial u_{i}}} \rbrack}}$

[0042] In the above equation, log denotes natural logarithm and E_(u)represents expected value based on p(x|u). A density on K for Lebesguemeasure du of Jeffreys prior distribution is represented as w_(J)(u),and obtained by:${w_{J}(u)} = \frac{\sqrt{\det \quad ( {J(u)} )}}{C_{J}(K)}$${{{where}\quad {C_{J}(K)}}\overset{def}{=}{\int_{K}\sqrt{\det \quad ( {J(u)} )\quad {u}}}},$

[0043] that is, C_(J)(K) is representative of a normalization constant.

[0044] Next, Jeffreys procedure proposed by B. Clarke and A. R. Barronwill be explained. Their method responds to inputs x^(n) and x_(n+1) andproduces outputs given by:${p_{J}( {x_{n + 1}x^{n}} )}\overset{def}{=}\frac{\int_{K}{p\quad ( {x^{n + 1}u} )\quad {w_{K}(u)}\quad {u}}}{\int_{K}{p\quad ( {x^{n}u} )\quad {w_{K}(u)}\quad {u}}}$

[0045] Next, redundancy which is used as a performance measure whichthey employ is introduced. Let q(x_(t)|x^(t−1)) represent an outputcorresponding to an input (x^(t−1),x_(t)) obtained a certain procedureq. Herein, it is assumed that each x_(t) (t=1,2, . . . ,n+1) is a randomvariable following a certain p(·|u)(u∈U_(c)⊂U). Redundancy for u of q isdetermined by:${R_{n}( {q,u} )}\overset{def}{=}{\sum\limits_{t = 1}^{n}{E_{u}\lbrack {\log \quad \frac{p\quad ( {x_{t}u} )}{q\quad ( {x_{t}x^{t - 1}} )}} \rbrack}}$

[0046] This may be referred to as cumulative Kullback-Leiblerdivergence. The value is always non-negative and means that theperformance of q becomes more excellent as the value becomes small. Inparticular, this index is often used in the context of data compression.Also, it is noted that the redundancy may be rewritten as follows.${R_{n}( {q,u} )} = {E_{u}\lbrack {\log \quad \frac{p\quad ( {x^{t}u} )}{q\quad ( x^{n} )}} \rbrack}$

[0047] Optimality of Jeffreys procedure proposed by B. Clarke and A. R.Barron is realized when the following equation is true.${R_{n}( {p_{J},u} )} = {{\frac{d}{2}\quad \log \quad \frac{n}{2\quad \pi \quad e}} + {\log \quad {C_{J}(K)}} + {o(1)}}$

[0048] Herein, the value of o(1) nears zero as n increases in amount.This asymptotic equation uniformly holds for all u (u∈K0) when let K₀ beany compact set which included K^(o) (i.e. K₀⊂K^(o)).

[0049] Because$\sup\limits_{u\quad \in \quad K}\quad {R_{n}( {q,u} )}$

[0050] is larger than or equal to the above value R_(n)(p_(J),u)whenever q takes any value, the asymptotic equation is optimum. That is,the following equation holds.${\inf\limits_{q}\quad \sup\limits_{u\quad \in \quad K}\quad {R_{n}( {q,n} )}} = {{\frac{d}{2}\quad \log \quad \frac{n}{2\quad \pi \quad e}} + {\log \quad {C_{J}(K)}} + {o(1)}}$

[0051] For the above relationship, p_(J) may be represented asasymptotically minimax for redundancy.

[0052] Next, logarithmic regret is introduced. It is assumed that asequence of data x^(n) is given. Logarithmic regret for the data x^(n)of q with respect to probability model S is defined by followingequation.${r\quad ( {q,x^{n}} )}\overset{def}{=}{\sum\limits_{t = 1}^{n}{\log \quad \frac{p\quad ( {x_{i}{\hat{u}\quad (n)}} )}{q\quad ( {x_{i}x^{t - 1}} )}}}$

[0053] Where, û(n) is maximum likelihood estimation value of u oncondition that the x^(n) is given. That is, û(n) is defined as follows.

[0054]${\hat{u}\quad (n)}\overset{def}{=}{\arg \quad {\max\limits_{u}{p( {x^{n}u} )}}}$

[0055] Like in the case of redundancy, the logarithmic regret isrepresented another way as follows.${r\quad ( {q,x^{n}} )} = {\log \quad \frac{p\quad ( {x^{n}{\hat{u}\quad (n)}} )}{q( x^{n} )}}$

[0056] In this point, when S is assumed to be an exponential family,following equation (1) holds. $\begin{matrix}{{r\quad ( {p_{J},x^{n}} )} = {{\frac{d}{2}\quad \log \quad \frac{n}{{2\quad \pi}\quad}} + {\log \quad {C_{J}(K)}} + {o(1)}}} & (1)\end{matrix}$

[0057] Exponential family is a model which can be represented by thefollowing equation.

S={p(x|θ)=exp(θ·x−ψ(θ)):θ∈Θ}

[0058] According to practice for notation of exponential family, θ maybe used as parameter instead of u. θ is referred to as natural parameteror θ- coordinates in exponential family. More detail description is madein L. Brown, “Fundamentals of statistical exponential families”,Institute of Statistics, 1986.

[0059] Asymptotic equation (1) uniformly holds for all x^(n) whichsatisfies û(n)∈K₀. Like in the case of redundancy, if the followingequation is true, q has property of minimax for logarithmic regret.However, when S does not belong to exponential family, the aboveasymptotic equation for Jeffreys procedure is not true. Instead, it canbe proved that the following formula holds.${\sup\limits_{x^{n}:{{\hat{u}{(n)}} \in K_{0}}}r\quad ( {p_{J},x^{n}} )} > {\inf\limits_{q}\sup\limits_{x^{n}:{{\hat{u}{(n)}} \in K}}r\quad ( {q,x^{n}} )}$

[0060] Taking the above into consideration, some modifications arerequired Here, one of solutions is explained. First, empirical Fisherinformation Ĵ is introduced and is given by:${\hat{J}( {xu} )}\overset{def}{=}{- \frac{{\partial^{2}\log}\quad {p( {xu} )}}{{\partial u_{j}}{\partial u_{i}}}}$

[0061] Furthermore, a definition is added as follows.${\hat{J}( {x^{n}u} )}\overset{def}{=}{\frac{1}{n}{\sum\limits_{t = 1}^{n}{\hat{J}\quad ( {x_{t}u} )}}}$

[0062] In this case, the following equation holds.${\hat{J}( {x^{n}u} )} = {{- \frac{1}{n}}\quad \frac{{\partial^{2}\log}\quad {p( {x^{n}u} )}}{{\partial u_{j}}{\partial u_{i}}}}$

[0063] Using the definitions of J gives:

J(u)=E _(u) [Ĵ(x ^(n) |u)]

[0064] Next, a random variable s is defined by:${{s( {xu} )}\overset{def}{=}{{\hat{J}( {xu} )} - {J(u)}}},$

[0065] where s is representative of a d-dimensional square matrix. Likein the case of the definition of Ĵ(x^(n)|u), s(x^(n)|u) is defined by:${s( {x^{n}u} )} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}{s( {x_{i}u} )}}}$

[0066] Let ν be representative of the d-dimensional square matrix. Inthis event, a family of new probability density is defined by:${\overset{\_}{p}( {{xu},v} )}\overset{def}{=}{{p( {xu} )}{\exp ( {{v \cdot {s( {xu} )}} - {\psi ( {u,v} )}} )}}$${{where}\quad {v \cdot {s( {xu} )}}}\overset{def}{=}{\sum\limits_{ij}{v_{ij}{s_{ij}( {xu} )}\quad {and}}}$$\begin{matrix}{{\psi ( {u,v} )}\overset{def}{=}\quad {\log {\int{{p( {xu} )}{\exp ( {v \cdot {s( {xu} )}} )}{v( {x} )}}}}} \\{= \quad {\log \quad {{E_{u}\lbrack {\exp ( {v \cdot {s( {xu} )}} )} \rbrack}.}}}\end{matrix}$

[0067] In this case, it is noted that {overscore (p)}(x^(n)|u, ν) isrepresented by:

{overscore (p)}(x ^(n) |u, ν)=p(x ^(n) |u)exp(n(ν·s(x ^(n) |u)−ψ(u, ν)))

[0068] Next,${V_{B}\overset{def}{=}\{ {{v:{\forall{i{\forall j}}}},{{v_{ij}} \leq B}} \}},$

[0069] and S is expanded into {overscore (S)} on the assumption that Bis representative of a certain positive constant and ψ(u,v) is finitefor u∈u, ν∈V_(B).

{overscore (S)}={{overscore (p)}(·|u, ν):u∈u, ν∈V _(B)}

[0070] {overscore (S)} thus obtained by expansion of S is referred to asexponential fiber bundle for S. In this case, the meaning of adjective“exponential” indicates that s(x|u) has the same direction asexponential curvature of S. More detail description is made in“Differential geometry in statistical inference”, Institute ofMathematical Statistics, Chapter 1, 1987.

[0071] Let ρ(u) be prior density on u and mixture density m be definedby:

[0072]${m( x^{n} )}\overset{def}{=}{\int{{\overset{\_}{p}( {{x^{n}u},v} )}{\rho (u)}{u}{{v}/( {2B} )^{d^{2}}}}}$

[0073] Herein, a range of integral for ν is V_(B). Also it is noted that(2B)^(d) ² is Lebesgue volume of V_(B).

[0074] Mixture is constructed by combining the following equation (2)with p_(J).

[0075] $\begin{matrix}{{{q^{(ɛ)}( x^{n} )}\overset{def}{=}{{( {1 - ɛ_{n}} ){p_{j}( x^{n} )}} + {ɛ_{n} \cdot {m( x^{n} )}}}},} & (2)\end{matrix}$

[0076] where 0<ε_(n)<½. For q in the above equation, it is assumed thatvalue of ε_(n) is decreased according to an amount of n and thefollowing inequality (3) holds. $\begin{matrix}{{\forall n},{ɛ_{n} \geq \frac{1}{n^{l}}}} & (3)\end{matrix}$

[0077] In the formula (3), 1 is representative of a certain positivenumber. On the basis of these assumptions, it is proved that q^((ε))asymptotically becomes minimax as the value of n increases.

[0078] This shows that, when q(x_(t)|x^(t−1)) is calculated not only byusing the mixture for S like in the general Bayes procedure but also byslightly combining the mixture m(x^(n)) for enlarged class, thecalculation brings about a good result with respect to the logarithmicregret.

[0079] When S belongs to the model referred to as a curved exponentialfamily, the procedure can be more simplified. This is specified by thecase where S belongs to a smooth subspace of the exponential family T.More specifically, on the assumption that T is a {overscore(d)}-dimensional exponential family given by

(T={p(x|θ)=exp(θ∫x−ψ(θ)):θ∈Θ⊂

^({overscore (d)})}),

[0080] S is represented on the condition of (d<{overscore (d)}) by:

S={p _(c)(x|u)=p(x|φ(u)):u∈U⊂

^(d)}

[0081] where φ is a smooth function characterized in that u|→θ. Forexample, if χ is a finite set, any smooth model becomes a curvedexponential family. Although the curved exponential family has highgenerality in comparison with the exponential family, it is notgeneralized as compared with the general smooth model class. More detaildescription is made by Shunichi Amari in “Differential-geometricalmethods in statistics”, Lecture notes in Statistics, Springer-Verlag.

[0082] Under these circumstances, {overscore (S)} (exponential fiberbundle of S) is coincident with T by the first-order approximation.Therefore, mixture in exponential family T in which S is included can beused instead of the mixture in exponential fiber bundle. That is, it canbe proved like in the above that q^((ε)) becomes minimax on theassumption given by: m(x^(n)) = ∫_(Θ^(′))p(x^(n)θ)ρ(θ)θ

[0083] In the above equation, Θ′ represents a set including{Θ:θ=φ(u),u∈U}, ρ represents a smooth prior distribution density on theΘ′.

[0084] In addition, in the case of the curved exponential family,calculation of Fisher information J(u) becomes easy. That is, in thiscase, an expected value can be determined without any operation by thefollowing equation (4). $\begin{matrix}{{J_{ij}(u)} = {{\sum\limits_{\alpha = 1}^{\overset{\_}{d}}{\sum\limits_{\beta = 1}^{\overset{\_}{d}}{\frac{\partial{\varphi_{\alpha}(u)}}{\partial u_{i}}\frac{\partial{\varphi_{\beta}(u)}}{\partial u_{j}}\frac{\partial^{2}{\psi (\theta)}}{{\partial\theta_{\alpha}}{\partial\theta_{\beta}}}}}}_{\theta = {\varphi {(u)}}}}} & (4)\end{matrix}$

[0085] This is because the Fisher information of θ in the exponentialfamily T is given by:$\frac{\partial^{2}{\psi (\theta)}}{{\partial\theta_{\alpha}}{\partial\theta_{\beta}}}$

[0086] In the previous description, it has thus far been explained thatlogarithmic regret which is used as the performance measure forsequential prediction issue can be minimized by combining the mixture onthe exponential fiber bundle with the Bayes mixture. This isadvantageous even if the logarithmic loss is used as the performancemeasure for non-sequential prediction issue. This is because suggestionis made about the fact that decreasing a value of the following formulaconcerned with the logarithmic loss results in a decrease of each termin the equation, namely, log(1/(q(x_(t)|x^(t−1)))). The values oflog(1/(q(x_(t)|x^(t−1)))) are referred to as the logarithmic loss. Now,the formula in question is given by:$\sum\limits_{t = 1}^{n}{\log \quad \frac{p( {x_{t}{\hat{u}(n)}} )}{q( {x_{t}x^{t - 1}} )}}$

[0087] It is an object of the present invention to provide a statisticalestimation method which is improved by using the above describedtechniques.

[0088] Herein, description will be schematically made about firstthrough eighth embodiments according to the present invention so as tofacilitate understanding of the present invention.

[0089] In the first embodiment according to the present invention,calculation is carried out in connection with the modified Bayes mixtureprobability density. To this end, an output generated from a devicewhich determines the Bayes mixture probability density on S is combinedwith an output generated from a device which calculates the Bayesmixture probability density on the exponential fiber bundle.

[0090] The second embodiment according to the present invention which isbasically similar to the first embodiment is featured by using theJeffreys prior distribution on calculating the Bayes mixture probabilitydensity on S.

[0091] The third embodiment according to the present invention isoperable in a manner similar to the first embodiment except thatoperation is simplified when S is curved exponential family. Suchsimplification can be accomplished by utilizing the property of S.

[0092] The fourth embodiment according to the present invention which isbasically similar to the third embodiment is featured by using theJeffreys prior distribution in the device which determines the Bayesmixture probability density on S.

[0093] The fifth through the eighth embodiments according to the presentinvention are featured by calculating prediction probability density bythe use of the devices according to the first through the fourthembodiments according to the invention, respectively.

[0094] Next, description will be made in detail about the first throughthe eighth embodiments according to the present invention with referenceto the accompanying drawings.

[0095] Referring to FIG. 1, a device according to the first embodimentof the invention is operated in a following order or sequence.

[0096] (1) Inputs x^(n) are provided to and stored into a probabilitydensity calculator shown by the block 11 in FIG. 1.

[0097] (2) Next, a Bayes mixture calculator shown by the block 12 inFIG. 1 calculates p(x^(n)|u) for various values of u by the use of theprobability density calculator 11 and also calculates approximationvalues of the Bayes mixture (given by p_(w)(x^(n))=∫p(x^(n)|u)w(u)du) byusing the previous calculation results p(x^(n)|u). Thereafter, the Bayesmixture calculator 12 sends the approximation values to a whole mixturecalculator shown by the block 14 in FIG. 1.

[0098] (3) An enlarged mixture calculator shown by the block 13 in FIG.1 calculates p(x^(n)|u) for various values of u and p(x|u) for variousvalues of both x and u in cooperation with the probability densitycalculator and calculates J(u) and Ĵ(x^(n)|u) for various values of u bythe use of previous calculation results p(x^(n)|u) and p(x|u). Further,using these results, the enlarged mixture calculator 13 calculates{overscore (p)}(x^(n)|u, ν) for various values of v and u, andcalculates approximation values of Bayes mixture m(x^(n))=∫{overscore(p)}(x^(n)|u,ν)ρ(u)dudν/B^(d) ² by the use of the previous calculationresults {overscore (p)}(x^(n)|u,ν) and sends the approximation values tothe whole mixture calculator 14.

[0099] (4) The whole mixture calculator 14 calculates the mixtureq^((ε))(x^(n))=1−ε)p_(w)(x^(n))+ε·m(x^(n)) for a predetermined smallvalue of ε on the basis of the values of two Bayes mixtures which havebeen stored and produces the mixture as an output.

[0100] Referring to FIG. 2, a device according to the second embodimentof the invention is basically similar in structure to the firstembodiment of the invention except that the device illustrated in FIG. 2utilizes a Jeffreys mixture calculator 22 instead of the Bayes mixturecalculator 12 used in FIG. 1. In FIG. 2, the device carries out nocalculation of the Bayes mixture ∫p(x^(n)|u)w(u)du but calculation ofthe Jeffreys mixture given by ∫p(x^(n)|u)w_(J)(u)du in accordance withthe above operation (2). That is, the Jeffreys mixture calculator 22calculates p(x^(n)|u) for various values of u and p(x|u) for variousvalues of x and u in cooperation with the probability density calculator21 and calculates J(u) for various values of u by using previouscalculation results p(x^(n)|u) and p(x|u). Subsequently, the Jeffreysmixture calculator 22 further calculates w_(J)(u) for various values ofu by the use of the previous calculation results to obtain approximationvalues of ∫p(x^(n)|u)w_(J)(u)du using w_(J)(u).

[0101] Referring to FIG. 3, a device according to the third embodimentof the invention is successively operated in order mentioned below.

[0102] (1) Inputs x^(n) are provided to and stored into a probabilitydensity calculator shown by 31 in FIG. 3.

[0103] (2) A Bayes mixture calculator shown by 32 in FIG. 3 calculatesp_(c)(x^(n)|u) for various values of u in cooperation with theprobability density calculator 31 and thereafter calculatesapproximation values of Bayes mixture p_(w)(x^(n))=∫p_(c)(x^(n)|u)w(u)duby using previous calculation results p_(c)(x^(n)|u). As a result, theapproximation values are sent from the Bayes mixture calculator 32 to astorage 34 which is operable as a part of a whole mixture calculator.

[0104] (3) An enlarged mixture calculator 33 in FIG. 3 calculatesp(x^(n)|θ) for various values of θ in cooperation with the probabilitydensity calculator 31 and calculates approximation values of Bayesmixture m(x^(n))=∫_(Θ′)p(x^(n)|θ)ρ(θ)dθ by using the previouscalculation results. The approximation values are sent from the enlargedmixture calculator 33 to the whole mixture calculator 34 in FIG. 3.

[0105] (4) The whole mixture calculator 34 calculates mixturesq^((ε))(x^(n))=(1−ε)p_(w)(x^(n))+ε·m(x^(n)) for a predetermined smallvalue of ε on the basis of the values of two Bayes mixtures which havebeen stored and produces the mixtures as outputs.

[0106] Referring to FIG. 4, a device according to the fourth embodimentof the invention is successively operated in order.

[0107] (1) Inputs x^(n) are provided to and stored into a probabilitydensity calculator shown by a block 41 in FIG. 4.

[0108] (2) A Jeffreys mixture calculator shown by 42 in FIG. 4calculates p_(c)(x^(n)|u) and w_(J)(u) for various values of u incooperation with the probability density calculator 41 and a Jeffreysprior distribution calculator 45 (which is designed according to theequation (4)). In addition, the Jeffreys mixture calculator 42calculates approximation values of Jeffreys mixturep_(J)(x^(n))=∫p_(c)(x^(n)|u)w_(J)(u)du by using the previous calculationresults p_(c)(x^(n)|u) and w_(J)(u), and sends the approximation valuesto a whole mixture calculator 44 in FIG. 4.

[0109] (3) An enlarged mixture calculator shown by 43 in FIG. 4calculates p(x^(n)|θ) for various values of θ in cooperation with theprobability density calculator 41 and obtain approximation values ofBayes mixture m(x^(n))=∫_(Θ′)p(x^(n)|θ)ρ(θ)dθ by using the previouscalculation results. The approximation values are sent from the enlargedmixture calculator 43 to the whole mixture calculator 44.

[0110] (4) The whole mixture calculator 44 calculates mixturesq^((ε))(x^(n))=(1−ε)p_(J)(x^(n))+ε·m(x^(n)) for a predetermined smallvalue of ε on the basis of the values of two Bayes mixtures which havebeen stored and produces the mixtures as outputs.

[0111] Referring to FIG. 5, a device according to each of the fifththrough eighth embodiments of the invention includes a joint probabilitydensity calculator 51. Herein, it is to be noted that the devicesillustrated in FIGS. 1 through 4 may be incorporated as the jointprobability density calculators 51 to the devices according to the fifththrough the eighth embodiments of the present invention, respectively.The device shown in FIG. 5 is operated in order mentioned below.

[0112] (1) Inputs x^(n) and x_(n+1) are provided to the jointprobability density calculator 51 in FIG. 5.

[0113] (2) The joint probability density calculator 51 calculatesq(x^(n)) and q(x^(n+1)) and sends the calculation results to a divider52 in FIG. 5.

[0114] (3) The divider calculates q(x_(n+1)|x^(n))=q(x^(n+1))/q(x^(n))by using the two joint probabilities sent from the joint probabilitydensity calculator 51.

[0115] According to the first embodiment of the invention, in regard toan issue of minimizing logarithmic regret for general probability modelS, it is possible to calculate more advantageous joint probabilitydistribution as compared with the conventional methods using traditionalBayes mixture joint probability on the S.

[0116] Furthermore, according to the second embodiment of the invention,in regard to an issue of minimizing logarithmic regret for generalprobability model S, it is possible to calculate more advantageous jointprobability distribution as compared with the methods using traditionalJeffreys mixture joint probability on the S.

[0117] Moreover, the third embodiment of the invention is advantageousin regard to an issue of minimizing logarithmic regret for curvedexponential family S in that the joint probability distribution iseffectively calculated as compared with the methods using traditionalBayes mixture joint probability on the S.

[0118] In addition, the fourth embodiment of the invention is effectivein connection with an issue of minimizing logarithmic regret for curvedexponential family S in that it is possible to calculate moreadvantageous joint probability distribution as compared with theconventional methods using traditional Jeffreys mixture jointprobability on the S.

[0119] Further, each of the fifth through the eighth embodiments of theinvention can effectively calculate the prediction probabilitydistribution in regard to a prediction issue using logarithmic loss asperformance measure, as compared with the conventional methods. Morespecifically, the fifth embodiment is more convenient than theconventional methods using traditional Bayes mixture joint probabilityon probability model S while the sixth embodiment is effective ascompared with the conventional methods using traditional Jeffreysmixture joint probability on probability model S. Likewise, the seventhembodiment of the invention is favorable in comparison with theconventional methods using traditional Bayes mixture joint probabilityon curved exponential family S while the eighth embodiment of theinvention is superior to the conventional methods using traditionalJeffreys mixture joint probability on curved exponential family S.

What is claimed is:
 1. A Bayes mixture density calculator operable inresponse to a sequence of vectors x^(n)=(x₁, x₂, . . . , x_(n)) selectedfrom a vector value set χ to produce a Bayes mixture density onoccurrence of the x^(n), comprising: a probability density calculator,supplied with a sequence of data x^(t) and a vector value parameter u,for calculating a probability density for the x^(t), p(x^(t)|u); a Bayesmixture calculator for calculating a first approximation value of aBayes mixture density p_(w)(x^(n)) on the basis of a prior distributionw(u) predetermined by the probability density calculator to produce thefirst approximation value; an enlarged mixture calculator forcalculating a second approximation value of a Bayes mixture m(x^(n)) onexponential fiber bundle in cooperation with the probability densitycalculator to produce the second approximation value; and a wholemixture calculator for calculating (1−ε)p_(w)(x^(n)+ε˜m(x^(n)) toproduce a calculation result by mixing the first approximation value ofthe Bayes mixture density p_(w)(x^(n)) with a part of the secondapproximation value of the Bayes mixture m(x^(n)) at a rate of 1−ε:ε toproduce the calculation result where ε is a value smaller than unity. 2.A Jeffreys mixture density calculator operable in response to a sequenceof vector x^(n)=(x₁, x₂, . . . , x_(n)) selected from a vector value setχ to produce a Bayes mixture density on occurrence of the x^(n),comprising: a probability density calculator responsive to a sequence ofdata x^(t) and a vector value parameter u for calculating a probabilitydensity p(x^(t)|u) for the x^(t); a Jeffreys mixture calculator forcalculating a first approximation value of a Bayes mixture densityp_(J)(x^(n)) based on a Jeffreys prior distribution w_(J)(u) incooperation with the probability density calculator to produce the firstapproximation value; an enlarged mixture calculator for calculating asecond approximation value of a Bayes mixture m(x^(n)) on exponentialfiber bundle in cooperation with the probability density calculator toproduce the second approximation value; and a whole mixture calculatorfor calculating (1−ε)p_(J)(x^(n))+ε·m(x^(n)) to produce a calculationresult by mixing the first approximation value of the Bayes mixturedensity p_(J)(x^(n)) with a part of the second approximation value ofthe Bayes mixture m(x^(n)) at a rate of 1−ε:ε to produce the calculationresult where ε is a value smaller than unity.
 3. A Bayes mixture densitycalculator operable in response to a sequence of vector x^(n)=(x₁, x₂, .. . , x_(n)) selected from a vector value set χ to produce a Bayesmixture density on occurrence of the x^(n), comprising: a probabilitydensity calculator responsive to a sequence of data x^(t) and a vectorvalue parameter u for outputting probability density p(x^(t)|u) for thex^(t) on curved exponential family; a Bayes mixture calculator forcalculating a first approximation value of a Bayes mixture densityp_(w)(x^(n)) on the basis of a prior distribution w(u) predetermined bythe probability density calculator to produce the first approximationvalue; an enlarged mixture calculator for calculating a secondapproximation value of a Bayes mixture m(x^(n)) on exponential familyincluding curved exponential family in cooperation with the probabilitydensity calculator to produce the second approximation value; and awhole mixture calculator for calculating (1−ε)p_(w)(x^(n))+ε·m(x^(n)) toproduce a calculation result by mixing the first approximation value ofthe Bayes mixture density p_(w)(x^(n)) with a part of the secondapproximation value of the Bayes mixture m(x^(n)) at a rate of 1−ε:ε toproduce the calculation result where ε is a value smaller than unity. 4.A Jeffreys mixture density calculator operable in response to a sequenceof vector x^(n)=(x₁, x₂, . . . , x_(n)) selected from a vector value setχ to produce a Bayes mixture density on occurrence of the x^(n),comprising: a probability density calculator responsive to a sequence ofdata x^(t) and a vector value parameter u for calculating probabilitydensity p(x^(t)|u) for the x^(t) on curved exponential family; aJeffreys mixture calculator for calculating a first approximation valueof a Bayes mixture density p_(J)(x^(n)) based on a Jeffreys priordistribution w_(J)(u) in cooperation with the probability densitycalculator to produce the first approximation value; an enlarged mixturecalculator for calculating a second approximation value of a Bayesmixture m(x^(n)) on exponential family including curved exponentialfamily in cooperation with the probability density calculator to producethe second approximation value; and a whole mixture calculator forcalculating (1−ε)p_(J)(x^(n))+ε·m(x^(n)) to produce a calculation resultby mixing the first approximation value of the Bayes mixture densityp_(J)(x^(n)) with a part of the second approximation value of the Bayesmixture m(x^(n)) at a ratio of 1−ε:ε to produce the calculation resultwhere ε is a value smaller than unity.
 5. A prediction probabilitydensity calculator operable in response to a sequence of vectorx^(n)=(x₁, x₂, . . . , x_(n)) selected from a vector value set χ andx_(n+1) to produce a prediction probability density on occurrence of thex_(n+1), comprising: a joint probability calculator structured by theBayes mixture density calculator claimed in claim 1 for calculating amodified Bayes mixture density q^((ε))(x^(n)) and q^((ε))(x^(n+1)) basedon predetermined prior distribution to produce first calculationresults; and a divider responsive to the calculation results forcalculating probability density q^((ε))(x^(n+1))/q^((ε))(x^(n)) toproduce a second calculation result with the first calculation resultskept intact.
 6. A prediction probability density calculator operable inresponse to a sequence of vector x^(n)=(x₁, x₂, . . . , x_(n)) selectedfrom a vector value set χ and x_(n+1) to produce a predictionprobability density on occurrence of the x_(n+1), comprising: a jointprobability calculator structured by the Jeffreys mixture densitycalculator claimed in claim 2 for calculating a modified Jeffreysmixture density q^((ε))(x^(n)) and q^((ε))(x^(n+1)) to produce firstcalculation results; and a divider response to the calculation resultsfor calculating a probability density q^((ε))(x^(n+1))/q^((ε))(x^(n)) toproduce a second calculation result with the first calculation resultskept intact.
 7. A prediction probability density calculator operable inresponse to a sequence of vector x^(n)=(x₁, x₂, . . . , x_(n)) selectedfrom a vector value set χ and x_(n+1) to produce a predictionprobability density on occurrence of the x_(n+1), comprising: a jointprobability calculator structured by the Bayes mixture densitycalculator claimed in claim 3 for calculating a modified Bayes mixturedensity q^((ε))(x^(n)) and q^((ε))(x^(n+1)) based on a predeterminedprior distribution to produce first calculation results; and a dividerresponsive to the calculation results for calculating a probabilitydensity q^((ε))(x^(n+1))/q^((ε))(x^(n)) to produce a second calculationresult with the first calculation results kept intact.
 8. A predictionprobability density calculator operable in response to a sequence ofvector x^(n)=(x₁, x₂, . . . , x_(n)) selected from a vector value set χand x_(n+1) to produce a prediction probability density on occurrence ofthe x_(n+1), comprising: a joint probability calculator structured bythe Jeffreys mixture density probability calculator claimed in claim 4for calculating a modified Jeffreys mixture density q^((ε))(x^(n)) andq^((ε))(x^(n+1)) to produce first calculation results; and a dividerresponsive to the calculation results for calculating a probabilitydensity q^((ε))(x^(n+1))/q^((ε))(x^(n)) to produce a second calculationresult with the first calculation results kept intact.
 9. A mixturedensity calculator operable in response to a sequence of data x^(n)=(x₁,x₂, . . . , x_(n)) to produce a mixture density on occurrence of thex^(n), comprising: receiving means for receiving the sequence of datax^(n); first calculation means for calculating a first Bayes mixturedensity on a hypothesis class; second calculation means for calculatinga second Bayes mixture density on an enlarged hypothesis class; andmeans for obtaining a modified Bayes mixture density for the x^(n) bymixing the first Bayes mixture density with the second Bayes mixturedensity in a predetermined proportion to produce the modified Bayesmixture density as said mixture density.
 10. A mixture densitycalculator as claimed in claim 9, wherein the first Bayes mixturedensity and the second Bayes mixture density are calculated by the useof a predetermined prior distribution.
 11. A mixture density calculatoras claimed in claim 9, in calculating the first Bayes mixture densityand second Bayes mixture density, uses Jeffreys prior distribution. 12.A mixture density calculator as claimed in claim 9, wherein the firstBayes mixture density and the second Bayes mixture density are mixedtogether at a rate of 1−ε:ε, where ε take a value smaller than unity.13. A mixture density calculator as claimed in claim 9, wherein thehypothesis class belongs to the curved exponential family.
 14. Aprediction probability density calculator operable in response to asequence of data x^(n)=(x₁, x₂, . . . , x_(n)) and a data x_(n+1) toproduce a prediction probability density on occurrence of the x_(n+1),comprising: receiving means for receiving the data x^(n) and x_(n+1);first calculating means for calculating first Bayes mixture densities,on a hypothesis class, for the sequence of data x^(n) and an sequence ofdata x^(n+1) which representing (x₁, x₂, . . . , x_(n), x_(n+1)); secondcalculating means for calculating second Bayes mixture densities, on anenlarged hypothesis class, for the x^(n) and the x^(n+1); means forobtaining modified Bayes mixture densities for the x_(n) and the x^(n+1)by mixing the first Bayes mixture densities for the x^(n) and thex^(n+1) with the second Bayes mixture densities for the x^(n) and thex^(n+1), in a predetermined proportion to produce the modified Bayesmixture density as said mixture density, respectively; and means forobtaining the prediction probability density by dividing the modifiedBayes mixture density for the x^(n+1) by the modified Bayes mixturedensity for the x^(n).
 15. A prediction probability density calculatoras claimed in claim 14, wherein the first Bayes mixture densities andthe second Bayes mixture densities are calculated by the use of apredetermined prior distribution.
 16. A prediction probability densitycalculator as claimed in claim 14, wherein the first Bayes mixturedensities and the second Bayes mixture densities are calculated by theuse of Jeffreys prior distribution.
 17. A prediction probability densitycalculator as claimed in claim 14, wherein the first Bayes mixturedensities and the second Bayes mixture densities are mixed together at arate of 1−ε:ε, where ε take a value smaller than unity.
 18. A predictionprobability density calculator as claimed in claim 14, wherein thehypothesis class belongs to curved exponential family.
 19. A methodoperable in response to a sequence of data x^(n)=(x₁, x₂, . . . , x_(n))to produce a mixture density on occurrence of the x^(n), comprising thesteps of: receiving the sequence of data x^(n); calculating a firstBayes mixture density on a hypothesis class; calculating a second Bayesmixture density on an enlarged hypothesis class; and obtaining amodified Bayes mixture density for the x^(n) by mixing the first Bayesmixture density with the second Bayes mixture density in a predeterminedproportion.
 20. A method operable in response to a sequence of datax^(n)=(x₁, x₂, . . . , x_(n)) and a data x_(n+1) to produce a predictionprobability density on occurrence of the x_(n+1), comprising the stepsof: receiving the data x^(n) and x_(n+1); and repeating, for eachsequence of data x^(n) and x^(n+1) representing (x₁, x₂, . . . , x_(n),x_(n+1)), the following first through third substeps of: (1) calculatinga first Bayes mixture density, on a hypothesis class, for the sequenceof data; (2) calculating a second Bayes mixture density, on an enlargedhypothesis class, for the data; and (3) obtaining a modified Bayesmixture density for the data by mixing the first Bayes mixture densitywith the second Bayes mixture density in a predetermined proportion; andobtaining the prediction probability density by dividing the modifiedBayes mixture density for the x^(n+1) by the modified Bayes mixturedensity for the x^(n).
 21. A computer readable medium which stores aprogram operable in response to an input a sequence of data x^(n)=(x₁,x₂, . . . , x_(n)) to produce a mixture density on occurrence of thex^(n), the program comprising the steps of: receiving the sequence ofdata x^(n); calculating a first Bayes mixture density on a hypothesisclass; calculating a second Bayes mixture density on an enlargedhypothesis class; and obtaining a modified Bayes mixture density for thex^(n) by mixing the first Bayes mixture density with the second Bayesmixture density in a predetermined proportion.
 22. A computer readablemedium which stores a program which is operable in response to an inputa sequence of data x^(n)=(x₁, x₂, . . . , x_(n)) and a data x_(n+1) toproduce a prediction probability density on occurrence of the x_(n+1),the program comprising the steps of: receiving the data x^(n) andx^(n+1); repeating, for each sequence of data x^(n) and x^(n+1) whichrepresenting (x₁, x₂, . . . , x_(n), x_(n+1)), the following substeps;(1) calculating a first Bayes mixture density, on a hypothesis class,for the sequence of data; (2) calculating a second Bayes mixturedensity, on an enlarged hypothesis class, for the data; (3) obtaining amodified Bayes mixture density for the data by mixing the first Bayesmixture density with the second Bayes mixture density, in apredetermined proportion; and obtaining the prediction probabilitydensity by dividing the modified Bayes mixture density for the x^(n+1)by the modified Bayes mixture density for the x^(n).